Question

What is the slope of the tangent line to the graph of a solution of$\mathit{y}\mathbf{\text{'}}\mathbf{=}\mathbf{6}\sqrt{\mathbf{y}}\mathbf{+}\mathbf{5}{\mathit{x}}^{\mathbf{3}}$that passes through (-1, 4)?

Short answer

The slope of the tangent line to the graph of a solution that passes through the point is 7.

Step-by-step solution

Define a derivative of the function

The sensitivity of the function is derived by the derivative of a function of a real variable for modifying the argument.

Calculus uses derivatives as a fundamental tool. When a derivative of a single-variable function exists at a given input value, it is the slope of the tangent line to the function's graph at that point.

Determine the differential equation for the height

Let the slope of the tangent line of the graph of $y=f\left(x\right)$at the point be y'.

And the slope of the tangent is,

$\begin{array}{c}{\left.y\text{'}\right|}_{(-1,4)}={\left.\left(6\sqrt{y}+5x^3\right)\right|}_{(-1,4)}\\ =6\sqrt{4}+5(-1{)}^3\\ =6\left(2\right)+5(-1)\\ =12-5\\ =7\end{array}$

Hence, the slope of the tangent line to the graph of a solution that passes through the point is 7.